By William Henry Besant
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During this quantity, that's devoted to H. Seifert, are papers in keeping with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.
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Extra resources for Conic Sections, Treated Geometrically
10. If SE be the perpendicular from the focus on the normal at P , shew that SE 2 = AN . SP. 11. The locus of the vertices of all parabolas, which have a common focus and a common tangent, is a circle. 12. Having given the focus, the length of the latus rectum, and a tangent, construct the parabola. 13. If P SP be a focal chord, and P N , P N the ordinates, shew that AN . AN = AS 2 . Shew also that the latus rectum is a mean proportional between the double ordinates. 14. The locus of the middle points of the focal chords of a parabola is another parabola.
Two equal parabolas have a common focus; and, from any point in the common tangent, another tangent is drawn to each; prove that these tangents are equidistant from the common focus. 112. Two parabolas have a common axis and vertex, and their concavities turned in opposite directions; the latus rectum of one is eight times that of the other; prove that the portion of a tangent to the former, intercepted between the common tangent and axis, is bisected by the latter. CHAPTER III. The Ellipse. Def.
2. Draw a tangent to a parabola, making a given angle with the axis. 3. If the tangent at P meet the tangent at the vertex in Y , AY 2 = AS . AN. 4. If the normal at P meet the axis in G, the focus is equidistant from the tangent at P and the straight line through G parallel to the tangent. 5. Given the focus, the position of the axis, and a tangent, construct the parabola. 6. Find the locus of the centre of a circle which touches a given straight line and a given circle. 7. Construct a parabola which has a given focus, and two given tangents.